Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients

TL;DR

This paper analytically solves a class of periodic Sturm-Liouville problems linked to Riccati equations, and introduces a novel numerical method to analyze the energy band structure of complex periodic potentials.

Contribution

It provides the first application of spectral parameter power series to periodic Sturm-Liouville problems and develops an efficient series representation of the Hill discriminant.

Findings

Analytical solutions for specific Riccati-related periodic problems.

Implementation of a new numerical approach for energy band analysis.

Derivation of a series representation for the Hill discriminant.

Abstract

We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically-coupled second-order differential equations. We solve analytically these parametric periodic problems along the positive real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for the first time in the investigation of the energy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and implements the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill discriminant based on Kravchenko's series